well, first of all the intagrand is an even function and so we just need to find the integral over then by parts gets us somewhere that we only need to find as you showed yourself. to find the value of i didn't use contour integration:

let first note that and so:

on the other hand, thus with solving this simple differential equation gives us
and therefore

it'd be nice to find an elementary proof of mine is not elementary yet because i used differentiating an integral with respect to a parameter.

I could 'feel' what your definition to an elementary proof is but perhaps much much sweat is needed for finding the elementary proof (Not sure is this true if you work in a room that the air-conditioner is on ).